Abstract：Integer partitions were first studied by Euler. In the first part of this talk, I shall introduce the hook length expansion technique and explain how to find old and new hook length formulas for integer partitions. In particular, we derive an expansion formula for the powers of the Euler Product in terms of hook lengths, which is discovered by Nekrasov-Okounkov and Westburg. Then, we prove that the Plancherel average of the even power sum of hook or content lengths is always a polynomial for certain classes of integer partitions, such as strict, doubled distinct and self-conjugate partitions. Finally, we mention the link with the limit shape conjecture for strict partition.
